By lex, on April 13th, 2007
NASCAR fan Max asked for a physics lesson on curvilinear path dive bombing for those who don’t wear horn-rimmed glasses. Which is a fair request, except for the fact that he, you know: Asked of it from me.
This is like getting a second opinion on your upcoming brain surgery from the local barber: He probably has an opinion to go along with his glancing familiarity with the issue at hand, but you don’t necessarily want to bet your life on it, one way or the other.
Anyway, here goes.
I may have mentioned before that where an airplane is pointing and where it is going are not always the same thing – in fact, it scarcely ever is. The longitudinal axis of the airframe – referred to as the armament data line (ADL) in this post – is where the machine is pointing. The flight path differs from the longitudinal axis as a function of angle of attack, which is itself the angular measure of the wingfoil’s mean chord line from the air mass it’s moving through. This is very obvious at very slow speeds and relatively high angles of attack – imagine the nose up attitude on the airliner you see that is descending for landing – and proportionately smaller for high speed/low-AOA aircraft. Like those in a dive bombing run.
But – and pace to those who think that size does matter – smaller does not mean “nothing,” and especially in dumb bombing precision counts. In the graphic below (all of these are linked to larger files), the line “b’ ” is the desired bomb delivery angle to the target at vertex “C”, while “h” represents the pilot’s sight picture at roll-in. The angle between them is angle of attack, and the line “a” is the track over terrain. If I had thought to put it in their before closing PowerPoint, there ought to be an “a’ ” between the target and the aim-off point referred to the aim-off distance referenced in the “Wire” post. (Note: This is actually a somewhat simplified picture, since bomb ballistics from the release point are non-linear, and the bomb release aimpoint itself is depressed over a hundred mils from the ADL – Graphic updated to show true relationship.)
The relationship between b and h is the cosine of the reference angle, definitionally. That is to say, the cosine of the angle at vertex A is adjacent (b) over hypotenuse (h,) while b’ is the angle subtended from the AOD and ADL (not to scale).
Gluttons for punishment may read more:
If we just maintained this relationship through the run – fixated on the aim-off vertex B, in other words, we’d find ourselves with a sight picture like this, and a weapons release angle different than what we’d planned (never mind for the instant the ballistics and accuracy of the release itself):
Intuitively we can see that we’re now aimed at the initial visual aim point, but what we really wanted to do was to put the bomb at it’s release point with reference to the target. It’s pretty clear that 1) a dumb bomb is going to land long of the target or 2) a smart bomb is going to land more vertically than we had intended (even if its guidance mechanism can hack the turn). Sub optimal.
One alternative would be to roll-in at an initial offset and let the target “creep up” to the desired release point, thus:
If that looks hard to reliably do, go to the head of the class. It is.
Another solution is to use curvilinear path tracking. To bunt the jet, in other words, to a g conforming to the cosine of the dive angle.
For a 45 degree dive, that’s 0.9g.
That sight picture ends up looking something like this:
In this case we have our planned dive angle and aim-off distance, with the red line conforming to the actual flight path required to freeze the picture all the way to release. That flight path curves a bit, so it requires us to unload (or bunt) the jet to something slightly less than 1.0g.
When I was first taught dive bombing in the A-4, we actually used a combination of straight-path and curvilinear dive bombing – and for computed bombing, we still sort of do – but it requires a great deal of practice to be worth a damn at it since there are so many different things you have to calculate in your head while the windscreen fills up with target at the better part of 500mph.
For instance (if I recall correctly) for a 45 degree bombing run, the planned release altitude was designed at being exactly at altitude, on dive angle and at airspeed with the aiming “pipper” right over the bullseye (for a no wind solution – there’s always wind). For every 10 knots fast you were supposed to release 100′ high, and for every 1 degree shallow you’d release 100′ low (for two examples) and of course these were changing dynamically as you attempted to wrestle the jet into parameters with altimeter unspooling like the clock on the wall of a 1950′s time-traveler sci fi, and by the way, in combat people were very likely shooting at you. And I don’t believe I mentioned it, but there are very few targets which are actually at sea level (the altimeter’s reference) so you’ve got to do that math as well.
Thus the investment in computed bombing and eventually, smart bombs.